Coloring Random Graphs

13 years 4 months ago

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An equitable coloring of a graph is a proper vertex coloring such that the sizes of any two color classes diﬀer by at most one. The least positive integer k for which there exists an equitable coloring of a graph G with k colors is said to be the equitable chromatic number of G and is denoted by χ=(G). The least positive integer k such that for any k′ ≥ k there exists an equitable coloring of a graph G with k′ colors is said to be the equitable chromatic threshold of G and is denoted by χ∗ =(G). In this paper we investigate the asymptotic behavior of these coloring parameters in the probability space G(n, p) of random graphs. We prove that if n−1/5+ǫ < p < 0.99 for some 0 < ǫ, then almost surely χ(G(n, p)) ≤ χ=(G(n, p)) = (1 + o(1))χ(G(n, p)) holds (where χ(G(n, p)) is the ordinary chromatic number of G(n, p)). We also show that there exists a constant C such that if C/n < p < 0.99, then almost surely χ(G(n, p)) ≤ χ=(G(n, p)) ≤ (2 + o(1))χ(G...

Michael Krivelevich, Benny Sudakov

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